High order numerical methods to two dimensional delta function integrals in level set methods

In this paper we design and analyze a class of high order numerical methods to delta function integrals appearing in level set methods in two dimensional case. The methods comprise approximating the mesh cell restrictions of the delta function integral. In each mesh cell the two dimensional delta function integral can be rewritten as a one dimensional ordinary integral with the smooth integrand being a one dimensional delta function integral, and thus is approximated by applying standard one dimensional high order numerical quadratures and high order numerical methods to one dimensional delta function integrals proposed in [X. Wen, High order numerical methods to a type of delta function integrals, J. Comput. Phys. 226 (2007) 1952–1967]. We establish error estimates for the method which show that the method can achieve any desired accuracy by assigning the corresponding accuracy to the sub-algorithms and has better accuracy under an assumption on the zero level set of the level set function which holds generally. Numerical examples are presented showing that the second to fourth order methods implemented in this paper achieve or exceed the expected accuracy and demonstrating the advantage of using our high order numerical methods.     

Two methods for discretizing a delta function supported on a level set

This paper presents two new methods for discretizing a Dirac delta function which is concentrated on the zero level set of a smooth function u: Rⁿ  R. The function u is only known at the discrete set of points belonging to a regular mesh covering Rⁿ. These two methods are used to approximate integrals over the manifold defined by the level set. Both methods are conceptually simple and easy to implement. We present the results of numerical experiments indicating that as the mesh size h goes to zero, the rate of convergence is at least O(h) for the first method, and O(h²) for the second method. We perform a limited analysis of the proposed algorithms, including a proof of convergence for both methods.     

The numerical approximation of a delta function with application to level set methods

It is shown that a discrete delta function can be constructed using a technique developed by Anita Mayo [The fast solution of Poisson’s and the biharmonic equations on irregular regions, SIAM J. Sci. Comput. 21 (1984) 285–299] for the numerical solution of elliptic equations with discontinuous source terms. This delta function is concentrated on the zero level set of a continuous function. In two space dimensions, this corresponds to a line and a surface in three space dimensions. Delta functions that are first and second order accurate are formulated in both two and three dimensions in terms of a level set function. The numerical implementation of these delta functions achieves the expected order of accuracy.     

Maximizing band gaps in two-dimensional photonic crystals by using level set methods

The optimal design of photonic band gaps for two-dimensional square lattices is considered. We use the level set method to represent the interface between two materials with two different dielectric constants. The interface is moved by a generalized gradient ascent method. The biggest gap of GaAs in air that we found is 0.4418 for TM (transverse magnetic field) and 0.2104 for TE (transverse electric field).     

A conservative level set method suitable for variable-order approximations and unstructured meshes

This paper presents a formulation for free-surface computations capable of handling complex phenomena, such as wave breaking, without excessive mass loss or smearing of the interface. The formulation is suitable for discretizations using finite elements of any topology and order, or other approaches such as isogeometric and finite volume methods. Furthermore, the approach builds on standard level set tools and can therefore be used to augment existing implementations of level set methods with discrete conservation properties. Implementations of the method are tested on several difficult two- and three-dimensional problems, including two incompressible air/water flow problems with available experimental results. Linear and quadratic approximations on unstructured tetrahedral and trilinear approximations on hexahedral meshes were tested. Global conservation and agreement with experiments as well as computations by other researchers are obtained.     

The constrained reinitialization equation for level set methods

Based on the constrained reinitialization scheme [D. Hartmann, M. Meinke, W. Schröder, Differential equation based constrained reinitialization for level set methods, J. Comput. Phys. 227 (2008) 6821–6845] a new constrained reinitialization equation incorporating a forcing term is introduced. Two formulations for high-order constrained reinitialization (HCR) are presented combining the simplicity and generality of the original reinitialization equation [M. Sussman, P. Smereka, S. Osher, A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys. 114 (1994) 146–159] in terms of high-order standard discretization and the accuracy of the constrained reinitialization scheme in terms of interface displacement. The novel HCR schemes represent simple extensions of standard implementations of the original reinitialization equation. The results evidence the significantly increased accuracy and robustness of the novel schemes.     

Shape and topology optimization for electrothermomechanical microactuators using level set methods

In this short note, a shape and topology optimization method is presented for multiphysics actuators including geometrically nonlinear modeling based on an implicit free boundary parameterization method. A level set model is established to describe structural design boundary by embedding it into the zero level set of a higher-dimensional level set function. The compactly supported radial basis functions (CSRBF) are introduced to parameterize the implicit level set surface with a high level of accuracy and smoothness. The original more difficult shape and topology optimization driven by the Hamilton–Jacobi partial differential equation (PDE) is transferred into a relatively easier parametric (size) optimization, to which many well-founded optimization algorithms can be applied. Thus the structural optimization is transformed to a numerical process that describes the design as a sequence of motions of the design boundaries by updating the expansion coefficients of the size optimization. Two widely studied examples are chosen to demonstrate the effectiveness of the proposed method.     

A variational approach to path planning in three dimensions using level set methods

In this paper we extend the two-dimensional methods set forth in [T. Cecil, D. Marthaler, A variational approach to search and path planning using level set methods, UCLA CAM Report, 04-61, 2004], proposing a variational approach to a path planning problem in three dimensions using a level set framework. After defining an energy integral over the path, we use gradient flow on the defined energy and evolve the entire path until a locally optimal steady state is reached. We follow the framework for motion of curves in three dimensions set forth in [P. Burchard, L.-T. Cheng, B. Merriman, S. Osher, Motion of curves in three spatial dimensions using a level set approach, J. Comput. Phys. 170(2) (2001) 720–741], modified appropriately to take into account that we allow for paths with positive, varying widths. Applications of this method extend to robotic motion and visibility problems, for example. Numerical methods and algorithms are given, and examples are presented.     

Variational piecewise constant level set methods for shape optimization of a two-density drum

We apply the piecewise constant level set method to a class of eigenvalue related two-phase shape optimization problems. Based on the augmented Lagrangian method and the Lagrange multiplier approach, we propose three effective variational methods for the constrained optimization problem. The corresponding gradient-type algorithms are detailed. The first Uzawa-type algorithm having applied to shape optimization in the literature is proven to be effective for our model, but it lacks stability and accuracy in satisfying the geometry constraint during the iteration. The two other novel algorithms we propose can overcome this limitation and satisfy the geometry constraint very accurately at each iteration. Moreover, they are both highly initial independent and more robust than the first algorithm. Without penalty parameters, the last projection Lagrangian algorithm has less severe restriction on the time step than the first two algorithms. Numerical results for various instances are presented and compared with those obtained by level set methods. The comparisons show effectiveness, efficiency and robustness of our methods. We expect our promising algorithms to be applied to other shape optimization and multiphase problems.     

A novel approach to segmentation and measurement of medical image using level set methods

The study proposes a novel approach for segmentation and visualization plus value-added surface area and volume measurements for brain medical image analysis. The proposed method contains edge detection and Bayesian based level set segmentation, surface and volume rendering, and surface area and volume measurements for 3D objects of interest (i.e., brain tumor, brain tissue, or whole brain).Two extensions based on edge detection and Bayesian level set are first used to segment 3D objects. Ray casting and a modified marching cubes algorithm are then adopted to facilitate volume and surface visualization of medical-image dataset. To provide physicians with more useful information for diagnosis, the surface area and volume of an examined 3D object are calculated by the techniques of linear algebra and surface integration. Experiment results are finally reported in terms of 3D object extraction, surface and volume rendering, and surface area and volume measurements for medical image analysis.     

A level set simulation of dendritic solidification with combined features of front-tracking and fixed-domain methods

A method combining features of front-tracking methods and fixed-domain methods is presented to model dendritic solidification of pure materials. To explicitly track the interface growth and shape of the solidifying crystals, a front-tracking approach based on the level set method is implemented. To easily model the heat and momentum transport, a fixed-domain method is implemented assuming a diffused freezing front where the liquid fraction is defined in terms of the level set function. The fixed-domain approach, by avoiding the explicit application of essential boundary conditions on the freezing front, leads to an energy conserving methodology that is not sensitive to the mesh size. To compute the freezing front morphology, an extended Stefan condition is considered. Applications to several classical Stefan problems and two- and three-dimensional crystal growth of pure materials in an undercooled melt including the effects of melt flow are considered. The computed results agree very well with available analytical solutions as well as with results obtained using front-tracking techniques and the phase-field method.     

Reconstructing absorption and diffusion shape profiles in optical tomography by a level set technique

A shape reconstruction algorithm for optical tomography is introduced that uses a level-set formulation for the shapes. Evolution laws based on gradient directions for a cost functional are derived for two different level-set functions, one describing the absorption and one the diffusion parameter, as well as for the parameter values inside these shapes. Numerical experiments are presented in 2D that show that the new method is able to simultaneously recover shapes and contrast values of absorbing and scattering objects embedded in a moderately heterogeneous background medium from simulated noisy data.     

Domain growth in ternary fluids: A level set approach

We analyze phase separation in ternary systems in the asymptotic hydrodynamic regime when the volume fractions and concentrations are constant. The multiphase Navier-Stokes equations are solved using a level set method. A new projection method was developed to treat multiple junctions for systems with more than two phases. It is found that surface tension ratios can alter the growth mechanism of a minority phase in the presence of two majority phases. When the minority phase wets the interface of the majority phases the domain growth rate of all three phases is initially similar to that of a symmetric binary fluid but slows down at later times.     

On the use of the resting potential and level set methods for identifying ischemic heart disease: An inverse problem

The electrical activity in the heart is modeled by a complex, nonlinear, fully coupled system of differential equations. Several scientists have studied how this model, referred to as the bidomain model, can be modified to incorporate the effect of heart infarctions on simulated ECG (electrocardiogram) recordings.We are concerned with the associated inverse problem; how can we use ECG recordings and mathematical models to identify the position, size and shape of heart infarctions? Due to the extreme CPU efforts needed to solve the bidomain equations, this model, in its full complexity, is not well-suited for this kind of problems. In this paper we show how biological knowledge about the resting potential in the heart and level set techniques can be combined to derive a suitable stationary model, expressed in terms of an elliptic PDE, for such applications. This approach leads to a nonlinear ill-posed minimization problem, which we propose to regularize and solve with a simple iterative scheme.Finally, our theoretical findings are illuminated through a series of computer simulations for an experimental setup involving a realistic heart in torso geometry. More specifically, experiments with synthetic ECG recordings, produced by solving the bidomain model, indicate that our method manages to identify the physical characteristics of the ischemic region(s) in the heart. Furthermore, the ill-posed nature of this inverse problem is explored, i.e. several quantitative issues of our scheme are explored.     

A level set projection model of lipid vesicles in general flows

A new numerical method to model the dynamic behavior of lipid vesicles under general flows is presented. A gradient-augmented level set method is used to model the membrane motion. To enforce the volume- and surface-incompressibility constraints a four-step projection method is developed to integrate the full Navier–Stokes equations. This scheme is implemented on an adaptive non-graded Cartesian grid. Convergence results are presented, along with sample two-dimensional results of vesicles under various flow conditions.     

A multi-phase level set framework for source reconstruction in bioluminescence tomography

We propose a novel multi-phase level set algorithm for solving the inverse problem of bioluminescence tomography. The distribution of unknown interior source is considered as piecewise constant and represented by using multiple level set functions. The localization of interior bioluminescence source is implemented by tracing the evolution of level set function. An alternate search scheme is incorporated to ensure the global optimal of reconstruction. Both numerical and physical experiments are performed to evaluate the developed level set reconstruction method. Reconstruction results show that the proposed method can stably resolve the interior source of bioluminescence tomography.     

The nonsinglet spin-dependent structure function evolution by Laplace and characteristics methods

We evaluate the non-singlet spin-dependent structure function g₁^NS at leading order (LO) and next-to-leading order (NLO) by using the Laplace-transform technique and method of characteristics and also obtain its first moment at NLO. The polarized non-singlet structure function results are compared with the data from HERMES (A. Airapetian et al., Phys. Rev. D 75, 012007 (2007)) and E143 (K. Abe et al. (E143 Collab.), Phys. Rev. D 58, 112003 (1998)) at LO and NLO analyses and the first-moment the result at NLO is compared with the result of the NLO GRSV2000 fit. Considering the solution, this method is valid at low- and large-x regions.     

A hybrid level set method for modelling detonation and combustion problems in complex geometries

We present an accurate and fast wave tracking method that uses parametric representations of tracked fronts, combined with modifications of level set methods that use narrow bands. Our strategy generates accurate computations of the front curvature and other geometric properties of the front. We introduce data structures that can store discrete representations of the location of the moving fronts and boundaries, as well as the corresponding level set fields, that are designed to reduce computational overhead and memory storage. We present an algorithm we call stack sweeping to efficiently sort and store data that is used to represent orientable fronts. Our implementation features two reciprocal procedures, a forward ‘front parameterization’ that constructs a parameterization of a front given a level set field and a backward ‘field construction’ that constructs an approximation of the signed normal distance to the front, given a parameterized representation of the front. These reciprocal procedures are used to achieve and maintain high spatial accuracy. Close to the front, precise computation of the normal distance is carried out by requiring that displacement vectors from grid points to the front be along a normal direction. For front curves in two dimensions, a cubic interpolation scheme is used, and G ¹ surface parameterization based on triangular patches is used for the three-dimensional implementation to compute the distances from grid points near the front. To demonstrate this new, high accuracy method we present validations and show examples of combustion-like applications that include detonation shock dynamics, material interface motions in a compressible multi-material simulation and the Stephan problem associated with dendrite solidification.